At the cathode,
ions accept electrons from the external circuit and become lithium
atoms, which plate out as a metal on the surface of the electrode:

At the anode,
ions give up electrons to the external circuit and combine into
neutral
molecules, which bubble away as a gas:

Such an arrangement for passing current through a molten salt is
called an electrolysis cell ("electro-lysis" meaning
"breaking down with electricity").
Electrolysis is one of the best ways of preparing pure metals such
as aluminum, which is electrolyzed commercially from a melt of aluminum
oxide ore.
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In the context of the present discussion, electrolysis cells themselves
are not as important as is the essential idea that current in a
molten salt is carried by the migration of positive and negative
ions.
Current is carried in the same way in a solution of LiF or any other
salt in water: The hydrated positive and negative ions move in opposite
directions.
Salt crystals do not conduct electricity, because the ions are locked
in a crystalline lattice and can not move.
Hall-Heroult cells in early aluminium production
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